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Sketch the graph of a function f so that... • There is a removable discontinuity at x 3 • There is a jump discontinuity at x = 1 lim f(x) = 3 lim f(x)= 4 X-2 X-5* lim f(x)= メ→-5 lim f(x)- 2

Sketch the graph of a function f so that... • There is a removable discontinuity at x 3 • There is a jump discontinuity at x = 1 lim f(x) = 3 lim f(x)= 4 X-2 X-5* lim f(x)= メ→-5 lim f(x)- 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title: Understanding Discontinuities and Limits in Functions** **Introduction:** In this lesson, we will explore how to sketch the graph of a function \( f \) that embodies various types of discontinuities and limit behaviors. Understanding these concepts is crucial for analyzing more advanced functions in calculus. **Graph Requirements:** 1. **Removable Discontinuity at \( x = 3 \):** - A removable discontinuity is typically a "hole" in the graph where the function is not defined, but the limit exists. 2. **Jump Discontinuity at \( x = 1 \):** - A jump discontinuity occurs when the left-hand and right-hand limits at a point differ, causing a sudden "jump" in function values. **Limit Conditions:** - \( \lim_{{x \to -2}} f(x) = 3 \): The function approaches a value of 3 as \( x \) approaches -2. - \( \lim_{{x \to 5^+}} f(x) = 4 \): The function approaches a value of 4 as \( x \) approaches 5 from the right (positive side). - \( \lim_{{x \to 5^-}} f(x) = -\infty \): As \( x \) approaches 5 from the left (negative side), the function goes to negative infinity, indicating a vertical asymptote at \( x = 5 \). - \( \lim_{{x \to -\infty}} f(x) = 2 \): As \( x \) goes to negative infinity, the function value approaches 2, indicating a horizontal asymptote. **Graph Construction:** Based on these criteria, your graph should reflect: - A small open circle at \( x = 3 \) (removable discontinuity). - A distinct vertical gap at \( x = 1 \) (jump discontinuity). - The specified asymptotic and limit behaviors as \( x \) approaches the given values. **Conclusion:** Creating such a graph reinforces the understanding of discontinuities and limits. This exercise is vital for visualizing function behavior and preparing for more complex calculus topics.
Transcribed Image Text:**Title: Understanding Discontinuities and Limits in Functions** **Introduction:** In this lesson, we will explore how to sketch the graph of a function \( f \) that embodies various types of discontinuities and limit behaviors. Understanding these concepts is crucial for analyzing more advanced functions in calculus. **Graph Requirements:** 1. **Removable Discontinuity at \( x = 3 \):** - A removable discontinuity is typically a "hole" in the graph where the function is not defined, but the limit exists. 2. **Jump Discontinuity at \( x = 1 \):** - A jump discontinuity occurs when the left-hand and right-hand limits at a point differ, causing a sudden "jump" in function values. **Limit Conditions:** - \( \lim_{{x \to -2}} f(x) = 3 \): The function approaches a value of 3 as \( x \) approaches -2. - \( \lim_{{x \to 5^+}} f(x) = 4 \): The function approaches a value of 4 as \( x \) approaches 5 from the right (positive side). - \( \lim_{{x \to 5^-}} f(x) = -\infty \): As \( x \) approaches 5 from the left (negative side), the function goes to negative infinity, indicating a vertical asymptote at \( x = 5 \). - \( \lim_{{x \to -\infty}} f(x) = 2 \): As \( x \) goes to negative infinity, the function value approaches 2, indicating a horizontal asymptote. **Graph Construction:** Based on these criteria, your graph should reflect: - A small open circle at \( x = 3 \) (removable discontinuity). - A distinct vertical gap at \( x = 1 \) (jump discontinuity). - The specified asymptotic and limit behaviors as \( x \) approaches the given values. **Conclusion:** Creating such a graph reinforces the understanding of discontinuities and limits. This exercise is vital for visualizing function behavior and preparing for more complex calculus topics.
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